Collective Dislocation Phenomena

in Non-equilibrium Systems

M.-Carmen MiguelUniversity of Barcelona

Paolo Moretti, Jérôme Weiss,

Michael Zaiser, Stefano Zapperi

KITP 2005

Outline� Plastic deformation of crystals

� Plastic avalanches in ice single crystals

� Plastic avalanches in 2D dislocation dynamics models

� Statistical properties under constant stress (creep tests)

� Dislocation jamming transition

� Aging like phenomena

� Slip heterogeneity & surface morphology � On microscopic scales …

� The vortex lattice in thin superconducting films: Dislocation dynamics & Plastic flow in quasi-2D

� Vortex polycrystals: Columnar grain growth

� Quenched disorder—Grain boundary pinning/depinning

� Experimental data

� Numerical results: critical current, hysteresis—Metastability

� Conclusions

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Under the action of constant stress:

SECONDARY: Stationary TERTIARY: Recovery.

Usually ends in fracture

PLASTIC

STRAIN-RATE

TIME

PRIMARY

Power law: t-2/3 “Andrade’s creep”

Time laws of plastic creep

Same behavior observed in several materials!

Continuum plasticity: Plastic deformation resembles a

smooth flow process, quasi-laminar flow of dislocations

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Compression creep test on an ice single crystal.Piezoelectric transducers are fixed directly on the sample

TRANSPARENT

Defectsinterference

Cracks

Plastic flow of ice single crystals

Nature 410, 667 (2001)

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ACOUSTIC EMISSION (AE) FROM COLLECTIVEACOUSTIC EMISSION (AE) FROM COLLECTIVE

DISLOCATION MOTIONDISLOCATION MOTION

CREEP

COMPRESSION

Small

shear stress on

the basal planes

Deforms by slip of dislocations

on the basal planes along a

preferred direction

SINGLE SLIP

ACOUSTIC

EMISSION

SUDDEN CHANGES

OF

INELASTIC STRAIN

Ice ����

c axis

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Energy distribution of acoustic events P(E)

05.06.1

)(

±=

= −

E

EEEP

ττττ

ττττ

Power law

distribution

Applied Stress0.58 MPa -1.64MPa

Resolved shear stress0.03 MPa - 0.086 MPa

Bursts of activity: Collective dislocation rearrangements

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∑≠

=nm

nmnn rbx )(r

& σµ

Dislocation dynamics model� 2D cross section & point-like edge dislocations.

� Random initial configuration (ρ0 = N/L2b2 ~ 1 – 5 %)

� Burgers vectors +b or –b with equal probability (single slip).

( )( ) nm

22

nm

2

nm

2

nm

2

nmnmmnm

r

1

yx

)y(xxDbyx,σ ∝

+

−=

� Relaxation=Numerically solve (adaptive-step-size fifth

order Runge-Kutta) the coupled

over-damped equations of motion:

with long range interactions until the system reaches a

still configuration (Ewald sums).

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� Annihilation of dislocation pairs at short distances rnm ~ 2b:

Remaining dislocation density (ρ ~ 0.5 – 1 %)

+= ∑

≠

e

nm

nmnn rbx σσµ )(r

&

IF HIGH STRESS σσσσ > σσσσ* Activation threshold value

∑∝•

n

nnvbt )(γ

� Apply constant external stress σe to relaxed configurations

� Creation of new dislocation pairs: Frank-Read sources (FRS’s).

� Creep dynamics: Strain rate vs. time

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Andrade’s & Linear Creep

BOX SIZE 100b x 100b

t-2/3

Slow power law

relaxation of the“average” strain-rate

for almost all the

time span

ttj βασγ ++= 3/1

0/

Andrade’s & Exponential

decrease toward no strain-rate

Low stress dynamics� Without creation of dislocations.

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σe=0.0125

N<v2> ~ Kinetic energy at the points where we have dislocations

Red one

Before After

While“Intermittent” signals:

Bursts of activity

Three individual runs: Microscopic behavior

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Outside the wall

Before

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Fast dislocations collaborating in the collective rearrangement

)( σvv >

While

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Inside the wallN has remained constant in this case!

After

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BOX SIZE 300b x 300bCrossover to linear regime(crossover time gets shorter with r)

3/1)( tt ∝γγγγ

Same results hold:� With creation of new dislocations.

� For different system sizes & multiplication rates r.

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SINGULAR RESPONSE “AVALANCHES’’

In the stationary state

� Rearrangements of dislocation

structures (Dipoles, walls, cells, etc.)

� Annihilation of dislocation pairs.

� Creation of new dislocations in FRS’s.

� Power law distributions for intermediate size

avalanches ⇒ Absence of characteristic size.

� Exponential cutoffs for large size avalanches:

Cutoff ⇓ when σ ⇑.Nature 410, 667 (2001)

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Stress Shear

high

Dislocation

walls...

Dislocation

dipoles

low

SLOW FAST Dislocations

Sources of

self-induced

jamming!

Snapshots of the dynamics:

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2.08.1

)(

±=

= −

E

EEEP

ττττ

ττττ

“Acoustic” Energy

2VE =

Mean Velocity vs. time

∑=)(

)(tN

n

n

m

vtV

In good agreement with Acoustic Emission experiments!

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• Power law ⇒⇒⇒⇒ Absence of

characteristic correlation time

• Non-diffusive behavior

Correlations: Aftershock triggering in ice-quakes

Time correlations of the AE signal in the model

Average number of aftershockstriggered by dislocation avalanches

of magnitude M=log A.

NM~Mα

Average AE event rate after

avalanches of magnitude M=log A, per mainshock and per unit of time.

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INTERMITTENCY, SLOW RELAXATION, SCALING ⇒

PROXIMITY OF NON-EQUILIBRIUM CRITICAL POINTS

1/Dislocation density

StressσσσσY

MOVING PHASE

Temperature

NON EQUILIBRIUM JAMMING TRANSITION

JAMMEDPHASE

The jamming scenario

YIELD STRESS VALUE (ρρρρ,T)

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New framework to study a wide variety of AMORPHOUSsoft glassy materials: emulsions, colloidal suspensions,

foams, gels, etc.; and granular media.

Liu & Nagel, Nature 396, 21 (2001)

COMMON FEATURES

ησγ //)( 3/1

0 tCtjtJ ++==

naγσ &≈

15.0,/ ≤≤−≈= α

αγγση &&

� Slow dynamics governed by kinematical

constraints (density, geometry, etc.)

� Jamming ⇒ Suppression of temporal relaxation or the ability to explore the configurations space.

� Shear yielding ⇒ Liquid-like flow above the yield stress value.

� Creep compliance curves

� Non-linear rheology:

� Stress-strain relationships

� Viscosity

Andrade and viscous

flow

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Dislocation jamming in crystalsWe also find

� Broad region of slow dynamics due to kinematical constraints:

Metastable patterns ⇒ Andrade’s creep.

� Dislocation jamming below the yield stress threshold.

� Shear yielding above threshold: Primary, secondary regimes.

Exhaustive study of finite-size effects

Yield threshold value ?

( ) 06.20065.0−≈ eσγ&

Phys. Rev. Lett. 89, 165501 (2002).

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Dislocation jamming in crystalsWe also find

� Non-linear stress-strain relationships in the stationary

state.

3/1γσ &a≈

3/2/ −≈= γγση &&

3/2/ t≈= γση &

Cox- Merz rulen

aγσ &≈

Best fit in the steady state for the intermediate stress values

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�Aging-like behavior�Loading rate dependence

tw Waiting time (without loading) after a

sudden quench of random configurations

σσσσe=0.0125

(Reminds Struik´s aging experiment on a PVC sample (78))

Creep time

For the lower values of stress, the final

state depends on the stress rate of change

Aging phenomena

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Hurst exponent calculated from surface

profiles along the deformation direction

Slip steps on the surface of deformed Cu

manifest the collective motion of dislocations

Slip heterogeneity and surface morphology

Zaiser et al., PRL 93, 195507 (2005)

The heterogeneous distribution

of slip has also implications on the surface morphology of

plastically deformed crystals.

Self-affine roughness over several

orders of magnitude in scale

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On microscopic scales …

Work in progress: 2D colloidal crystals under different modes of deformation

Different microscopic mechanisms mediating plastic deformation,depending on boundary conditions,and/or deformation mode:

•Dislocation sequential motion•Grain boundary formation and gliding•Re-crystallization

Intermittent slip We also find:

128x20 - Snapshot at t=4000

Slip steps

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Vortex lattices in superconductors

� Magnetic decoration: Powerful tecnique to investigate the topology of the lattice.

� Topological defects:

Isolated dislocations,

dislocation dipoles, and

grain boundaries as in

conventional crystals.Delaunay triangulations ofthe magnetic flux line positions.

Mixed state of type II superconductors ⇒Magnetic field enters in the form of a flexible array of flux lines that, much like ordinary matter, can form crystalline, liquid, and glassy phases.

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Vortex phases & quenched disorder

� Experimentally accessibleto study dynamic phase behavior and/or phase transitions.

� Elastic properties easily tunable with magnetic field

� Dynamic phases: pinned, crystalline, plastic ⇒intermediate regime where the lattice is found to be quite disordered.

PLASTIC DEPINNING & PLASTIC FLOW•Nucleation & motion of dislocations.

•Yields a non linear response (I-V curves).

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Dislocations in the vortex array

Thin superconducting samples: Threading edge dislocations

∫ =L

bduBurgers vector b ⊥ dislocation axis

� 7:5 coordination � Peach-Koehler force,

τ×(σ⋅b), due to the elastic

shear stress field, σ~∇u, that is usually generated by quenched disorder, other defects, external

current distributions, etc.

u elastic displacement field in the vortex lattice

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Vortex polycrystals

� Magnetic decoration images, e.g.

after field cooling thin samples

NbSe2 at 36 Oe, and applying a

current J<Jc.

F. Pardo et al. PRL 78, 4633 (1997).

� Similar experiments (FC, ZFC, FCR):

Y. Fasano et al. PRB 66, 020512 (2002)

M. Menghini et al. PRL 90, 147001 (2003)

� Possible relevance in the “peak

effect” – Enhanced pinning strength.

Experimental evidence

Grain boundary separating crystalline columnar grains

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Grain growth

� Glide motion of grain boundaries (GB).

� Driving forces:

� Externally induced through current flow in the superconductor.

� Internally generated by the gradual ordering

process (residual stress) after a rapid quench.

� Tends to minimize the energy cost of GB,

� Average grain size R

� Straight grain boundaries

� Γ0 energy per unit area ~ Kb2/D

Increasing the grain size dR→ Energy gain

RV

E 0Γ∝

dRRV

dE2

0Γ∝

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Grain growth & Disorder (oxygen vacancies, impurities)

� Quenched disorder induces elastic deformations of GB’s.

c

gb

DR

σ0Γ

≈

� σc--Competition between non-local elasticity, disorder and driving forces in the framework of pinning theories.

Grain

Grain boundary network

Junction point

Energy gain (dR) ~ Work against the pinning forces on GB

Pinning forces

R

Dissipative work per

unit volume to move

all GB a small dR,

dRDR

bE c

p

σ≈

Limit grain size

PRL 92, 257004 (2004)

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Elastic properties of an ideal GB� An infinite & regularly spaced

low-angle grain boundary of edge dislocations piled-up

along the y axis and with Burgers vectors in the x

direction.

� Dislocation wandering within the slip plane

� Elastic Hamiltonian of the vortex lattice, i.e.

[ ]∫ ∂+⋅∇−+∇= 2

44

2

6611

2

66

3 )())(()(2/1 ucuccucrdH z

)),(()( iDzXzR ii =

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Elastic properties of an ideal GB

zyzy kcckckkM 664466, 2)( +≅

[ ]∑ ∫∫ −−+=yG

zyzyzyyz

BZ

ykQXkQXkGQM

dkdQ

D

bH ),(),(),(

222 2

2

ππ

π

In the harmonic approximation & taking into account appropriate constraints:

The long-distance (q→0) behavior of the kernel:

Or rescaling the y-axis by a factor

66

44

2

1

c

c

,)( , kKkkM zy ≅ with6644ccK =

A strongly non-local elastic kernel !

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Experimental results for grain size in NbMo

Weak pinning

I.V. Grigorieva, Sov. Phys. JETP 69, 194 (89).

Strong pinning

&Nonlocalelasticity

Local elasticity + strong pinning

PRL 92,

257004 (2004)

TheoreticalEstimates:

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Vortex polycrystal

� Initially, many dislocations annihilate and react with each other.

� Dislocations form GB’s.

� Growth of grains with various sizes and orientations.

� Structure gets pinned by disorder.

Numerical evidence

Vortex simulations of a rapid field cooling (FC) in the

stripe geometry—weak or

strong pinning. N5 vs. time

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Numerical analysis

N (up to 4128) vortices in a cell of size D (D 18,36,72λ).

Periodic boundary conditions.

Vortex mutual interactions:

~r-1 if r<λλλλ

K1(rij/λλλλ) Bessel function

λλλλ London penetration length ~e-r if r<λλλλ

Point-like pinning centers.

Initial configuration: Random configuration of vortex

lines.

� Keep track of vortex dynamics with or without current-ind.

forces.

� Calculate lattice topology, current thresholds, etc.

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Grain boundary pinning

Pinnedvortex &dislocationstructures.

Increasing density of vortex lines from a) till d)

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Depinning current with GB

� Grain boundaries also move in response to applied currents.

� Numerical data indicate:

� If GB after field cooling:

The depinning current Jc

increases, i.e. it is above

the single crystal value.

� The critical current Jc may

depend on the average

grain size, as the yield

stress for polycrystals

(Hall-Petch relation).

� Needs further study for different pinning regimes, cooling histories → average grain sizes, etc.

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Hysteretic behavior

� Commonly observed in FC critical current measurements.

� Above depinningJ>Jc(B):� Plastic flow mediated by

GB sliding.

� Metastability

� Upon ramping up and

down the driving current,

the number of dislocations

in GB decreases, and the

critical current decreases

from Jc1 to Jc2 .

In good agreement with transport experiments, i.e.

Z.L. Xiao et al., PRL 85, 3265 (2000),Z.L. Xiao et al., PRL 86, 2431 (2001).

Jc1Jc2

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Conclusions� Plastic avalanches in acoustic emission experiments and in 2D

dislocation dynamic models (Andrade’s creep, intermittency).� Power law distributions of plastic events.

� Aftershock triggering & slowly decaying time correlations.

� Dislocation induced jamming transition in 2D dislocation dynamic models.� Analogous rheology in soft amorphous materials ⇒ Jamming scenario.

� Spatio-temporal heterogeneity of slip ⇒ Self-affine rough surfaces

� After standard field-cooling protocols, we observe vortex polycrystalswhere grain growth is limited by grain boundary pinning.

� Elastic properties of grain boundaries: strongly non-local.

� Estimates for the depinning stress and the average grain size, under weak and strong pinning assumptions.

� Increase of the critical depinning current for vortex polycrystals.

� Hysteresis ⇒ Metastability.